Using the EXIF data, you should be able to use the following formula:
(Resolution in pixels/Focal plane resolution in dpi) X 25.4(mm/in)=size in mm

For my Canon 5DII, the horizintal and vertical numbers figure out to:
(5616p/3849.21ppi) X 25.4mm/in = 37.058mm
(3744p/3908.14ppi) X 25.4mm/in = 24.33mm

Once you have the horizontal and vertical measurements, you can calculate the diagonal measurement using the simple formula a² + b² = c² or c = √(a² + b²)

For the Canon 5DII, this figures out to: √(37.058² + 24.33²) = 44.33mm

As you can see, the result is slightly larger than the expected 36X24mm that would produce a 43.267mm diagonal. Canon publishes the sensor size as 36X24mm in sales literature, but specifies 35.8X23.9mm in technical literature for the 5DII. I got similar results for every camera I tried this with. Since the resolution reported is that of the image after demosaicing, the discrepancy can't be explained because of the extra edge pixels that are only used to interpolate the RGB values of the "effective" pixels used in the demosaiced image.

Rather than use the EXIF data, you could also consult a database such as can be found at DxO Mark. The sensor dimensions, in mm, of every camera they test is included in the specifications tab for each camera.

In general, FF cameras use sensors that are 36X24mm, APS-C cameras (other than Canon) use sensors that are 23.5X15.6mm, Canon APS-C sensors are 22.3X14.9mm. For a full list of sensor sizes used by almost every current manufacturer, see the chart near the end of http://en.wikipedia.org/wiki/Image_sensor_format