During my PhD, I subscribed to the RSS feeds of the journals I would regularly read (started with 6 of them, ended up with a dozen). I would skim through titles of all new articles, and read abstracts of those whose title drew my eye. I then found out that some journals (J. Chem. Phys. in that particular case) offer specific RSS feeds for each of their sections, in addition to the “whole journal” feed. That helped reducing the number of journals I was skimming through.
Now, after the end of my PhD, my research interest are broader, the number of journals I like to watch is larger but my time is more limited. This system didn't work anymore, and I set up a new system, which has worked well for a few years. I use bibliographic databases (SciFinder and Web of Science; but I'm sure Google Scholar and PubMed have the same features) to create publication and citation alerts. Here's what I have set:
Citation alerts for all my own papers: if someöne cites my work, there's a good chance I'll be interested in their paper. This one has two additional “strategic” bonuses: you get to keep an eye on your competition, and you can suggest newer work to other authors when relevant (“hi there, I saw your recent paper citing my 2008 article on X, I thought you might be interested on a new extension of this algorithm that we published this year”).
Publication alerts for major players in the field of interest: I have 10 to 20 of those, watching all papers these people publish.
Citation alerts for some seminal or high-impact papers by others in the field: a good way to see how a new idea is adopted/improved by the community. Those tend to trigger a massive number of cites, so you may want to get rid of them after some time. I have between 5 and 10 of those alerts at a given time.
The only drawback to this method: database updates tend to lag somewhat behind the RSS feeds of the journals themselves, so you get papers that are 2 to 8 weeks old.
In addition, use conferences to stay on top/catch up with the literature: