Merphy,
The T3500 motherboard may have the standard spacing for a conventional LGA1366 fan / heatsink, but I'm not absolutely certain as the T35500 has a proprietary fan, heatsink, and shroud. It's possible but not certain without knowing the spacing of the mounting screws. Somehow, in the plan view of the M/B, the holes look quite far apart and that is to accommodate the shroud that encloses the heatsink all around.
If you are going to be using an LGA1366 CPU on a Dell Precision T3500 motherboard, you may as well buy a T3500 and use the uprated steel /Copper heatsink. The T3500 can use any of the 4 and 6-core LGA1366, W3500, W3600, X5500, and X5600 CPU's. The 6-core X5690 3.47 /3.73GHz is the highest power user at 130W, can't be overclocked, and in the T3500 can run at full performance all the time using the uprated cooler. The nice feature of the arrangement ins that the T3500 fan /shroud, heatsink arrangement is very quiet. I have a T3500 with an X5677 4-core 3.47 /3.73GHz which is also 130W and on long 3D CAD and moderate rendering, it never falters. That system cost $53 + $24 shipping and was fully functional. I added $43 of RAM, plus $60 for the X5677 and plugged in a PERC 6'i RAID controller, 15K SAS drive, left over Quadro 4000 and it goes very well.
Also, there may or may not be certain features of the T3500 motherboard that will not match perfectly with a new case, and certain details won't fit. The one I can think of is the HD activity light has a 3-pin connector and on the PERC H310- which is a RAID controller newer than the T3500, the lead is 4-pin as does the 2013 and 2015 HP z420's.
If you intend to use LGA1366 Xeons in a new case, you might consider a new Supermicro LGA1366 motherboard and this will go into a standard new case and use a standard, high performance LGA1366 cooler. These will have a more modern slot arrangement- fewer PCI- and can use more RAM. The 24GB limit of the T3500 may or may not be enough for your use. The T3500 here has 12GB- filling all 6 slots with 2GB modules and now I wish I had the full 24GB. My other systems have 32GB and 48GB.
Also, you might consider a Precision T5500 which has an 875W power supply and 12 RAM slots. With a CPU/memory /fan riser, it can have two LGA Xeons.
By the way, both the T3500 and T5500 have two GPU slots- but not SLI capability.
I'm a big fan of LGA1366- the performance is still completely usable, the price is amazingly low, and the three LGA systems I've had have been 100% reliable hardware.
Another quick suggestion: If you use the T3500 motherboard, consider a PERC H310 or LSI 9240 controller and that will convert the disk system to 6GB/s. In the T5500 listed, the Passmark disk score changed from 1940 to 2649.
Cheers,
BambiBoom
LGA1366 Systems:
Dell Precision T3500 (2011) (Rev 2) Xeon X5677 4-core @ 3.46 / 3.73GHz > 12GB (6X 2GB) DDR3-1333 ECC > Quadro 4000 (2GB) > PERC 6/i + Seagate 300GB 15K SAS ST3300657SS + WD Black 500GB > 525W PSU> Windows 7 Professional 64-bit > 2X Dell 19" LCD
[Passmark system rating = 2751, CPU = 7236 / 2D= 658 / 3D=2020 / Mem= 1875 / Disk=1221]
Dell Precision T5500 (2011) (Revised) > 2X Xeon X5680 (6-core @ 3.33 / 3.6GHz), 48GB DDR3 1333 ECC Reg. > Quadro K2200 (4GB ) > PERC H310 / Samsung 840 250GB / WD RE4 Enterprise 1TB > M-Audio 192 sound card > Logitech z313 > 875W PSU > Windows 7 Professional 64> HP 2711x (27", 1920 X 1080)
[ Passmark system rating = 3844 / CPU = 15047 / 2D= 662 / 3D= 3550 / Mem= 1785 / Disk= 2649] (12.30.15)