First, perhaps consider your timing. It's the end of the semester on the day you posted your question. The faculty focus is on finishing up, grading and grant reporting, etc. Next are the holidays. Then, the first two weeks of the semester are about committees, budget issues and helping students set up. There are sweet spots before midterms, mid semester, and a few weeks before finals when faculty exhale. I suggest, if you have time, week 3 of the new semester. I realize for students this sounds odd; however, when there are a lot of students/classes as professor/teacher it's important for us to finish well and it's amazing how many letters come in on everything from car problems to job offers to health issues.
If you don't have time, here's a second strategy. I often receive reference requests that are so general/vague I have no idea where to start. When students send an "outline" of what they're looking for I can respond much quicker. For example, perhaps note the request, purpose, a reminder of particular projects, dates, assignments or papers in your follow up email. This sounds crass but sometimes a request comes a year or two after a class and I remember the student but not enough detail to support a recommendation. I don't need the letter written for me, I need something akin to a mind map to bring the student and course to mind in a way that adds value to the recommendation. Also, bluntly, from your course experience you probably know if the professor/teacher is organized to the level that you can count on him/her for a recommendation.
Lastly, a few ideas for students. A couple of days before most college applications are due is less than an ideal time for ask for recommendations due to the numbers of last minute requests received. Select faculty you worked with beyond attending class who will have something to say; I often receive requests from students who chose to remain somewhat anonymous in class and I don't know them well enough to offer something insightful. Faculty have posted office hours. An ideal reference situation is when a student comes in with a single page paper noting the dates, a few details of the course and reintroduces him/herself. Then we have a chance to chat for a few minutes. In that time I "travel" to our work and am excited to hear about the student's potential opportunity. This lends an enthusiasm to the reference.
I don't want this to sound like it's all about the faculty, I'm trying to nicely say you would never believe the requests that come in, I could write a post only on funny/sad/shocking email I receive every semester. There is great joy in seeing students transition to their next level of education and we're happy to write recommendations when we have the information to do so. Student consideration of timing, ideas/details provided and a referential conversation create a better opportunity for meaningful reference letters.
This advice goes beyond post grad; however, when I advised post grad students, up until last year, many of the issues were the same and I think the approach crosses education levels.