The Royals recently threw 2 consecutive complete games, one by Brian Bannister and one by Zack Greinke.
That makes them the 4th team to accomplish the feat since the beginning of the 2007 season:
Team StreakStart Streak End Games W L CG SHO SV IP H R ER BB SO HR ERA HBP WP BK Opponents
+-------+-----------+-----------+-----+------------------------------------------------------------------------------------------------------------------------+
KCR 2008-04-13 2008-04-14 2 2 0 2 0 0 18 8 2 1 2 7 0 0.50 1 0 0 MIN,SEA
CHW 2007-09-19 2007-09-20 2 1 1 2 1 0 17 11 3 3 1 10 0 1.59 2 0 0 KCR
PIT 2007-08-12 2007-08-13 2 2 0 2 1 0 18 8 1 1 1 8 0 0.50 0 0 0 SFG
LAA 2007-07-30 2007-07-31 2 1 1 2 1 0 17 15 2 2 4 12 0 1.06 0 0 0 SEA
Going back to 1997, there are 4 teams who have had 3 CGs in a row:
Team StreakStart Streak End Games W L CG SHO SV IP H R ER BB SO HR ERA HBP WP BK Opponents
+-------+-----------+-----------+-----+------------------------------------------------------------------------------------------------------------------------+
MIN 2004-07-05 2004-07-07 3 3 0 3 3 0 27 13 0 0 2 24 0 0.00 3 0 0 KCR
OAK 2000-09-08 2000-09-10 3 2 1 3 2 0 27 12 4 2 0 16 1 0.67 0 0 0 TBD
MIN 1999-07-30 1999-08-01 3 2 1 3 1 0 26 17 3 3 4 17 1 1.04 0 1 0 ANA
ATL 1998-06-14 1998-06-17 3 3 0 3 1 0 27 12 3 1 2 22 1 0.33 0 1 0 MON,FLA
And here are the leaders since 1956:
Team StreakStart Streak End Games W L CG SHO SV IP H R ER BB SO HR ERA HBP WP BK Opponents
+-------+-----------+-----------+-----+------------------------------------------------------------------------------------------------------------------------+
DET 1968-09-06 1968-09-19 12 10 2 12 3 0 108 81 20 20 18 98 11 1.67 0 1 0 MIN,CAL,OAK,NYY
OAK 1980-08-09 1980-08-17 9 6 3 9 0 0 86.1 63 20 17 12 43 9 1.77 1 0 0 SEA,MIN
CHW 1956-09-16 1956-09-23 8 5 3 8 0 0 75 55 22 18 12 34 4 2.16 0 1 1 WSH,NYY,KCA
I would bet that if we could go all the way back to 1901, there were lots of times when teams when 20, 30, or more games in a row with CGs, since there were periods back then when use of relief pitchers was very rare. It's ironic the type of reversal we've seen. Back in the very early 1900s, it was probably a significant pain in the butt for the manager if he had to bring in a relief pitcher.
These days, most teams have a relief pitcher who is accustomed to throwing every 2 days, or 2 out of every 3, etc, and having a few complete games in a row could potentially mess up such players. Of course, that problem is easy to fix--you can always have a relief guy throw a bullpen session to keep his arm fresh.