The situation you describe is pretty dire. At most US insitutions, an uncurved 61 is a D or F. If your mean is a D or F, and the median is below that (I think that's what you mean by leftward skew?) then most of the class is likely to fail.
I think the first thing you should ask yourself is whether the tests were fairly evaluating the goals you set for your students. If they were too hard for other reasons (not enough time given, numerical problems too difficult, poor directions), then you should fix these problems in future tests, which will have the effect of making your tests less difficult. You will probably also need to adjust your grading scheme to deal with the damaged tests, probably by weighting them less.
If you decide the tests were fair, then your students are not meeting the goals you set for them. I would encourage you to lower your goals. There might be a few cases were it is legitimate for an instructor to fail half a class. (A flight instructor shouldn't certify anyone who can't fly.) But these cases are really rare. Normally, I think that you should adjust the goals of the class to a level that most students can meet, so they are getting some benefit from the class. Lowering your goals for the class will have the effect of making exams more easy, but it will have the more important effect of making it relevant for your students.
I also think you should speak to your chair about whether this policy against curving is absolute. Solving the problems with your test and curricular goals, as described above, are the big tasks, but a curve could also be a helpful bandaid, and your administration doesn't want half your class to fail.
ADDED 4/24/2014 I just wanted to note that I wrote this answer before the OP clarified that this was a course with a common curriculum over several years and a consistent testing history, whose students had recently become unable (for whatever reason) to meet the previous standards. That's an incredibly messy situation, and I don't claim to have advice for it without actually knowing what larger forces are causing this. The only advice I would give strongly in that setting is that the OP should notify his higher ups of what is going on.