Triceps tendonitis is a very common problem (I have dealt with it on-and-off over the years). In my experience, it stems from one or both of two common issues:
1) Most lifters get a lot of incidental triceps stimulation. (Remember, every time you do any sort of a press or pullover, your triceps are getting worked.) So depending on their split, some lifters are hitting triceps hard 5+ days a week. Given this, it's not surprising the tendons get hinky after a while.
2) Most lifters do not take the time to warm up properly before placing inordinate stress on the triceps tendon. For example, I used to do heavy skullcrushers as my first triceps movement. This is a classic example of terrible programming, as this exercise is really stressful on the tendon (especially if done with heavy weights). Little wonder that I suffered triceps tendonitis for years.
I did a training cycle with John Meadows, and he really opened my eyes to the importance of appropriate intra-workout exercise programming. For any bodypart, the first exercise performed should be geared toward getting blood into the muscle without putting undo stress on the tendons and joints. To do this, an isolation exercise should be employed, light loads used, and (most importantly) the ROM is limited to the final third (ie, approaching the fully contracted position) of the movement. With respect to triceps, allow me to expand (Note: The following is based on something I added to the post entitled EyeDentist: How do you train?):
My favorite warm-up exercise for triceps is rope pushouts (not a typo--will explain below). Use a very light weight, and pull the rope handles down and OUT (this is key!) as hard as you can. The goal is to pull the rope hard enough to literally render it straight (ie, parallel to the floor). Hold this extreme-contracted position for a 4-count, then 'break' the peak contraction, slowly lower the weight stack a few inches, then it's right back to the max contracted position for another 4-count. I can't emphasize enough just how short the ROM is--the hands NEVER get close to one another, and other than the first and last reps, the movement has no vertical (ie, up and down) component. (This is why I call them rope pushOUTS, not pushDOWNS.) Done correctly, these are brutal, the pump is insane, and most importantly, they prepare the triceps tendon for heavier work. Do 3 sets, resting only 30 sec between each. The weight should be light enough to allow 15-25 reps, with at least 2-3 left in the tank (ie, don't go to failure).
If this exercise irritates your triceps tendons, you're gonna have to consider taking a six-week-or-so holiday from ALL triceps-related work.