There is a poor man's way if you own a videocamera, a tripod, a computer, and video editing software...on second thought, there are some pretty cheap radar guns out there.
Still, on the chance that you or your family already has the capability to take video and manipulate it on a computer, here is a way:
Set up a target at home plate and measure its exact distance from your release point. This will take some thought and a real measurement, rather than a crude estimate. If you pitch from 60'6", the distance from your release point to home plate will likely be in the neighborhood of 54' to 55' but each 1 foot of error in this measurement can affect your velocity calculations by up to +/- 2 mph.
Set up a vidcam on a tripod at 1st base (for RHP) with the zoom set so that the entire distance from pitcher's release point to home plate is captured on the view screen.
Start recording video, and go make a few pitches.
Download the video to your computer and open it in a videoediting program. In typical video editing programs you should be able to find the exact frame where the ball leaves your hand and the exact frame where the ball hits your target, or at least crosses through the plane of your target. At 29.97 frames per second each frame lasts for .0334 sec, so you can simply add the number of frames from release point to target and multiply by .0334 to get the total travel time in seconds. Let's say your fastball required 14 frames to go from your hand to the target:
Then, 14 x 0.0334 = 0.47 sec.
Let's say your distance is 54 feet:
Then, your FB is traveling at 54 ft/0.47 sec, or 115 ft/sec.
There are 5280 ft in a mile, and 3600 sec in an hour. So,
115 ft/sec x (3600sec/hr / 5280ft/mile) = 78.4 mph.
Unfortunately, the uncertainty in these calculations is somewhat problematic because you can't know the release point or arrival at target to better than +/- 17 milliseconds using standard video frame rates. In the 78 mph range this will translate into an uncertainty of +/- 1 mph.
Also, you have to realize this calculation gives the average velocity of your pitches, not the peak velocity. If you got 78 mph from this type of measurement on a given pitch you could estimate with reasonable accuracy that your peak speed is about 82 mph and the speed of this same pitch as it crosses home plate is about 74 mph.
It might be simpler to just buy a radar gun. The cheap ones like Bushnell (less than $100) are accurate to about +/- 1 mph and they can measure your peak speed (which is what most pitchers like to see). The main reason they are cheap is because they have a limited range of operation. That shouldn't matter if you simply want someone to gun your pitches in bullpen sessions.