This article was compiled from input from members of the community, most notably TuuS and appears reliable but is not officially validated by Lenovo.
The T61 / T61p was manufactured with one of two different families of Intel CPUs
If your system board came with a Merom and you install a Penryn, you'll get a false "thermal error" on boot. There is a 3rd party bios to block that error from posting, but it's unauthorized by Lenovo and should be considered experimental.
If your system board came with a Penryn, you can install either of the two CPUs. (These were introduced in the spring of 2008)
If your not sure of which CPU you have, load your bios and it will show you it's intel number. Mine is T9300 (which is a 2.5GB Penryn CPU with 6MB of cache). Go to Intel's website and search that number and towards the top of the page it will say something like "formerly Penryn" (I believe "formerly" is used because they are out of production).
Since Penryn is newer, your Merom boards won't be able to run the faster CPUs, so you can either try to find the fastest Merom CPU, or if you want to be daring you can reflash your bios with an unoffical firmware and run the Penryn. Although this has been done successfully with thinkpad enthousiats world-wide, it still won't give you full Penryn support.
The fastest Merom processor I know of is that a T61 will run is the T7800It runs at 2.6ghz and has 4MB cache (note that this is slightly faster then my 2.5ghz Penryn, but my 6MB of cache more then makes up for the slight difference in speed.
The fastest Penryn processor I know of is that the T61 will run is the T9500It runs at 2.6ghz and has 6MB cache.
There are faster processors, but I believe the T61 will not support one with a front size buss rate above 800mhz. *note, if anyone knows if this is possible please contact me and I'll edit this information.
This should fully explain all the technical details I've discovered during my investigation, but to make this topic more complete, I'll share a photographic guide showing how to remove and replace your CPU and cooling fan, and also addresses replacing your thermal paste (proper use of thermal paste is a MUST, to much and you could end up with a T61 paperweight, too little and you got the same problem. Basically when the heatsink is tightened down onto your CPU and GPU, it should spread to cover all the chip, with as little as possible overspread. I recommend Artic Silver #5, which is made of 99.9% silver filings. They also make a less expensive version that uses alumimum instead of silver, but given that T61 models with discrete graphics can generate a fair amount of heat under load, you don't want to get cheap paste to save yourself all of $5.
Buying a new fan should be considered too, especially if the bearings in the fan are loose or you have seen intermittent "fan error" messages on startup. You can spin the fan by hand and check to ensure it turns freely and that the blades don't wobble which would indicate play and worn bushings. Cleaning the blades and heatsinks while you have it out is important. You can buy the fan alone for $12 including shipping from many ebay sellers. The heatsink and fan can also be purchased as a unit, but your heat sink should last the life of the Laptop. There is also a conductive "pad" that transfers heat from your systemboard chipset. This can be reused, just make sure it is intact and don't allow it to get any dust or finger prints on it. I'd also recommend a can of compressed air to blow out the inside of the laptop. Do this before you remove your cpu, you don't want to be blowing debris into the socket.
Also be very careful not to bend any pins on the CPU, and inspect the new CPU carefully, a slightly bent pin could miss being detected and if you insert the cpu carelessly, you'll crush it. The CPU is keyed and the gold triangle stenciled on one corner should point to the top right corner of the socket when correctly oriented.