What are my options for hiding/removing service panels from the outside of a house?
The small black box may be the exterior electronic reader end of your water meter.  Trace the wire into the house and if it connects to your water meter then you probably can't do much about that box.  You may be able to have it relocated by your water utility.

The center grey box appears to be for phone lines and the one on the right looks like cable TV.  It should be safe to open both boxes to verify the wiring.  You should see a "customer access" screw on the right side of each.  I wouldn't recommend removing ether of them unless you don't plan on selling any time soon.  A potential buyer may well want broad band internet (DSL or cable modem).  I know it would be a deal breaker for me not to have high speed. ;-)

You may have better luck hiding the boxes and lines by painting them to match the brick or with placement of plants or wall hangings.  Finally, you may also consider using some cable ties to clean up the clutter without actually removing anything.  Just bundle the wires together when they run parallel.  Be aware that this could cause interference in the data lines though.