We report on the first 180 days of Rossi X-Ray Timing Explorer observations of the outburst of the black hole candidate IGR J17091-3624. This source exhibits a broad variety of complex light curve patterns including periods of strong flares alternating with quiet intervals. Similar patterns in the X-ray light curves have been seen in the (up to now) unique black hole system GRS 1915+105. In the context of the variability classes defined by Belloni et al. for GRS 1915+105, we find that IGR J17091-3624 shows the ν, ρ, α, λ, β, and μ classes as well as quiet periods which resemble the χ class, all occurring at 2-60 keV count rate levels which can be 10-50 times lower than observed in GRS 1915+105. The so-called ρ class "heartbeats" occur as fast as every few seconds and as slow as ~100 s, tracing a loop in the hardness-intensity diagram which resembles that previously seen in GRS 1915+105. However, while GRS 1915+105 traverses this loop clockwise, IGR J17091-3624 does so in the opposite sense. We briefly discuss our findings in the context of the models proposed for GRS 1915+105 and find that either all models requiring near Eddington luminosities for GRS 1915+105-like variability fail, or IGR J17091-3624 lies at a distance well in excess of 20 kpc, or it harbors one of the least massive black holes known (<3 M &sun;).