Maine hasn't elected a Republican Congressman since 1994, when James Longley beat Dennis "Duke" Dutremble 52%-48% for the seat vacated by Tom Andrews (D).  Mr. Andrews ran for the US Senate Seat after George Mitchell (D) decided not to seek reelection. He eventually lost to Olympia Snowe (R), who previously held the Congressional seat in the Second District.

Tom Allen beat Longley by ten points two years later. 

Only North Dakota, where Republicans last won in 1978), and Vermont and Hawaii, which last elected GOP Congressmen in 1988, have gone longer.  (The last two Republican Congressmen from Massachusetts lost in 1996.)

note: Edited to correct House race reference.