That likely will create some matchup problems late in games for manager John Gibbons, but the organization claims its not worried about the current predicament.
"I know, ideally, you have so many righties and so many lefties, [we're] just looking for guys who can get outs," general manager Alex Anthopoulos said.
"Sometimes people get so caught up in left-right, if you're a righty that gets left-handers out, who cares? We don't look at it that way, we're looking for guys who can get outs for us. Right, left, submarine, whatever it is, if you can get three outs, it's fine."
The situation could be rather short-term because prospect Aaron Sanchez is expected to receive a promotion soon. Sanchez, who is the club's top prospect, according to MLB.com, is a starter by trade, but recently transitioned to the bullpen in the Minor Leagues.
Sanchez would provide Gibbons with another option in relief, but until then, he'll have to make due with lefties Brad Mills, Brett Cecil, Aaron Loup and Rasmussen.
Gregor Chisholm is a reporter for MLB.com. Read his blog, North of the Border, and follow him on Twitter @gregorMLB. This story was not subject to the approval of Major League Baseball or its clubs.