The most obvious candidate is the Milankovitch cycles. Those cycles were the primary drivers of the ice ages (Hays et al. 1976), which were rather large changes in Earth's climate, with up to 4ºC differences in average global temperature between glacial and interglacial periods. The main reasons that rule out the Milankovitch cycles for this episode of global warming?
First, the Milankovitch cycles are too slow. At the end of the last ice age, global average temperature rose by around 3.5ºC over 8,000 years (Shakun et al. 2012). That's a rate of change of only 0.04375ºC per century. In contrast, between 1900 and 2000 AD, temperatures rose by 0.7ºC. That's a rate of change 16x faster than at the end of the last ice age. The current rate of change has accelerated as well. The rate over the past 30 years has been 1.6ºC per century, nearly 37x faster than the rate at the end of the last ice age.
Second, the Milankovitch cycles have been in cooling phases for the last 6,000 years with another 23,000 years of cooling to go before they switch into the next warming phase (Imbrie and Imbrie 1980). Up until 1900 AD, the Earth had been cooling in conjunction with the cooling Milankovitch cycles (Marcott et al. 2013). Not only is the current warming faster than the normal for the Milankovitch cycles, it's completely counter to them.
The next most obvious candidate is the sun. After all, the Stefan-Boltzmann law specifically shows that increases in solar output should warm the planet. Reconstructions of solar output and global temperature also show a good relationship between the two prior to AD 1970 (i.e. Usoskin et al. 2005). Unfortunately, the sun has gotten cooler over the last 30+ years:
Sunspot activity has fallen by more than half since 1957 and the current cycle is one of the quietest of the last century:
Multiple papers have been published in the last 10 years, all noting the disconnect between the sun and global average temperatures since the 1970s (Solanki et al. 2004, Usoskin et al. 2005, Scafetta and West 2006, Lockwood and Fröhlich 2007, Lean and Rind 2008, Benestad and Schmidt 2009, Erlykin et al. 2009, Huber and Knutti 2011). A cooler, less active sun is pumping out less energy whereas increasing global average temperature requires more energy. The only way the sun can be causing the current warming is if the law of conservation of energy has been repealed, along with the laws of thermodynamics.
Related to the idea that the sun is causing the current episode of global warming is the idea that cosmic rays are the cause. This was first proposed by Henrik Svensmark in 1998. His original idea was that cosmic rays influenced cloud formation by creating ions in the atmosphere which would attract water vapor, thereby forming clouds. A decrease in cosmic rays would lead to a drop in ion formation, thereby leading to fewer clouds and decreasing the Earth's albedo. That drop in albedo would allow more solar energy to reach the ground and be absorbed, thereby raising global temperatures.
Unfortunately for this proposed cycle, the amount of cosmic rays have increased since the 1980s (Lockwood and Fröhlich 2007), which should lead to a increase in clouds and an increase in Earth's albedo, thereby cooling the planet. Furthermore, the correlation between clouds and cosmic rays has completely broken down, with no correlation after 1994 (Laut 2003). Continuing that thread, Laken et al. (2012) found no correlation whatsoever between cosmic rays and clouds. Even more problematic? Svensmark's idea depends on clouds being a negative feedback on global temperature. More clouds = lower temperature, fewer clouds = higher temperature. Instead, recent data shows that clouds are either a small positive feedback on global temperature (Clement et al. 2009, Dessler 2010) or have little to no effect on global temperatures (Dessler 2011, Masters 2012).
There are several ocean cycles that one could choose. The first is El Niño/Southern Oscillation (ENSO), in which Pacific surface waters near the equator oscillate between warm and cool phases. When the oscillation is in a warm phase (El Niño), heat from the ocean is dumped into the atmosphere, warming surface temperatures around the planet while cooling the ocean slightly. Conversely, when the oscillation is in a cold phase (La Niña), the Pacific absorbs heat from the atmosphere, cooling the atmosphere while warming the ocean.
This raises the point that if global warming is due to the oceans, then we would expect the oceans to have cooled since the 1970s as the atmosphere warmed. Instead, the oceans have warmed along with the atmosphere (Murphy et al. 2009, Nuccitelli et al. 2012, Balmaseda et al. 2013). That's the first problem with the claim that ocean cycles are behind global warming. The other major problem? Oscillations by definition do not have any overall trend–they just oscillate around a zero point. So how can something that doesn't have a trend drive the trend in global temperatures?
Another ocean cycle is the Pacific Decadal Oscillation (PDO). I've dealt with this one before. Aside from the little problem that since 1900 the trend in atmospheric temperatures has risen faster when the PDO is in a cold phase, there's the problem of how the PDO is calculated. The way it's calculated is to take the average temperature of the North Pacific and subtract the global ocean average temperature (JISAO). What's left over is the PDO. All it shows is whether the North Pacific is warmer than the global average or colder than the average and by how much. It says nothing about whether or not the global average has risen over time.
Finally, there's the Atlantic Multidecadal Oscillation (AMO), which is calculated by taking the North Atlantic average temperature, calculating the trend in that data, then removing the trend. As with the PDO, the leftovers are the AMO. All it shows if if the North Atlantic is warmer than its own average or colder. It tells us nothing about whether or not that average has increased over time or whether or not the global average has increased over time.
There's nothing really to summarize. When you look closely at the claims that global warming is due to various natural cycles, the evidence shows otherwise. The next time someone claims that global warming is a natural cycle, ask that person to specify which natural cycle he or she has in mind–all the ones I know of don't work.