Collect assignments seat-to-seat rather than having students stack them up.

The typical way to collect assignments is to have students pile them in the front of the room.  This takes a frustratingly long time.

Especially if there's stapling to do.  Each student in turn, with a huge line behind them, painstakingly aligns papers just so before gently, gently, gently! applying the staple to just the right spot.  Then they check the staple the same way a six-year-old might self-satisfyingly inspect a booger, before reluctantly ceding the stapler to the next person in line.  And the process repeats.

You don't have the opportunity to shepherd the line or encourage haste, because four students, mouths flapping like baby birds begging for regurgitated fish, compete for your attention.  Three are offering excuses or asking for extensions; one merely has a question requiring an answer - right now! - about whether quantum wave functions influence Confucian philosophy.

At least three in a class of twenty will fail to turn in their assignment without calling for your attention.  Usually this is because they didn't do the assignment; but plenty of times it's because they simply spaced out during collection time.

"Do I have everyone's paper?" you ask.  The three students with delinquent assignments don't answer.  Yes, one is deliberately and passive-aggressively ignoring you, hoping you won't notice.  But one is honestly thinking you're asking about a different paper entirely.  The other one is thinking about sex and so didn't process what you said.

When you finally get around to going through the stack, you see that one person (most likely knowingly) left the second half of the assignment blank; this student and three others failed to put names on their work.

AARRGH!  What can you do?!?

You can't complain about slow stapling, or holler at folks to hurry the process up - you'll sound like an officious, nagging arse.  You can tell the hungry birdies to talk to you later, but they will still try to talk to you now.  You can call out individuals, asking whether they turned in their assignment; but you're using even more time, you might call out the wrong people, and you sound like an officious, nagging arse.  

And you can't wait until later to check who's turned in the work, because as soon as the students leave the room, they've forgotten about whatever they were supposed to turn in.  The assignment doesn't serve its purpose - getting students feedback on their understanding - if the students avoid showing you their understanding in the first place.

When I finally snapped, I decided to collect papers from each student's desk individually... and I've never gone back.

I bring the stapler with me so that *I* do all necessary stapling.  If someone doesn't have the work, I can charge an extension or make immediate arrangements for the student to get the work done at an alternate time.  I can glance at everyone's work as I pick it up, so that a substantially incomplete assignment gets the same treatment as a missing assignment.  

Most importantly, the collection process usually occurs while students are working on a quiz or on some sort of timed, quiet, individual work.  No one wants to tell a long story that invariably ends with "and therefore I don't have the assignment" - no, they want to get back to the quiz, so I just hear "extension, please."  If they forget to put their papers on their desk, they grab from their bookbag quickly and quietly when I come to their desk.  With an established routine and practice I can get a 15-20 person class collected in about two to three minutes.  

Note that this collection method isn't only for homework!  Any written assignment, any at all, can be checked this way.  I once saw master physics teacher Peggy Bertrand walk through the class putting her custom stamp on each student's problem log to mark their progress.  I've collected index cards with "check your neighbor" answers on them, shuffled, and picked one person to explain to the class; by collecting from each student's desk, I can be sure every card has a name and some sort of answer on it.

As always, I don't claim to have The One Right Way to collect assignments.  I'm merely sharing something that's worked for me...