You need to tell us more about your current monitor. If it is a flat screen CRT then you may find your eyestrain is removed by switching to a curved monitor (I'm in this boat, as flat CRT monitors make me feel like I will barf). Some people are the opposite, they find their eyes feel better when switching from a curved glass to a flat glass CRT.
Is there a lot of glare or reflections on your monitor. If so either block the light or try an anti-glare screen coating or screen cover. These will be $20 vs. the $500 new monitor cost.
Are you sitting too close or too far from the monitor? Are you sitting above or below the monitor?
I agree with some of the other comments. If you have >.25 dot pitch, then the monitor is crap, get a new one. Also, have you tried to reduce the brightness, contrast, or changed refresh rates?