FORTUNE -- Dear Annie: What can you say to coworkers who give you a hassle about taking time off for a family problem? The department where I work is under a lot of pressure to meet deadlines, and in the five years I've been here, I've done my share of the workload and then some. Now, however, I'm faced with having to move my widowed father, who lives halfway across the country and has Alzheimer's, into a nursing home. I also have to clean out the house where he's lived for 40-plus years and put it up for sale. I'll need at least two weeks, or possibly three, to do this and, believe me, I wish it wasn't necessary.

What's making the whole situation even worse is the attitude of coworkers and, to a lesser extent, my boss. People have said things to me like, "Isn't there someone else who can do this instead of you?" (there isn't) and "Can't it wait until we've met our July project deadline?" (no, unfortunately, it can't). But I feel like I'm letting the team down. How should I respond? -- Torn in Tacoma

Dear T.T.: As you probably already know, the Family and Medical Leave Act of 1993 entitles you to up to 12 weeks per year of time off — unpaid, but without losing your benefits, or your job -- to deal with a health-related issue, either yours or a close family member's. New Labor Department rules, which took effect about six weeks ago, require employers to put up posters in workplaces, spelling out what the FMLA says (and, not incidentally, reminding everybody that it exists).

But pointing out that you have a legal right to do what you need to do isn't likely to get you very far with your colleagues, or your boss. Moreover, you're far from the only one struggling with this. "A lack of support from peers has a tremendous effect on employees' feeling that they can legitimately take time off," notes Justin Boren, a communications professor at Santa Clara University and an expert on social structures within organizations.

Boren is co-author of a study in the April issue of Southern Communication Journal showing that "messages of peer resentment" often stop people from taking the family-leave time they're entitled to. Those messages -- some subtle, some not -- made most employees in Boren's research "feel guilty for taking their full complement of benefits, if it meant leaving their colleagues to 'pick up the slack.'"

Often, coworkers' resentment "stems from unstated expectations about performance that are embedded in the culture of the organization," he adds. "The stress of trying to balance work life and family life is really exacerbated when colleagues say you're letting the team down."

So, how can you defend yourself? "In every relationship at work, there is a quid pro quo," says Lois Frankel, head of Pasadena-based Corporate Coaching International, whose clients have included executives at Disney (DIS), Lockheed Martin (LMT), BP (BP), Amgen (AMGN), and many other big companies. "You have to give at least as much as you get."

A family crisis "often preoccupies us so completely that we forget to think much about how our absence will affect the people around us at work," Frankel observes. "That's where the resentment from colleagues comes in."

Before you leave to take care of your father, she suggests you sit down with your teammates, either as a group or one-on-one, and work out what you can contribute to the next project deadline before you take off, how reachable you'll be while you're away, and what additional work you'd be willing to handle when you get back.

"It's important to make clear that you don't intend to just leave everybody in the lurch," Frankel says. "So discuss with them ways to make their jobs less difficult and stressful despite your absence. It's also a good idea to explicitly offer to help fill in for anybody else who may need to take a leave in the future.

"Also, don't forget to manage up," she adds. "Have the same kind of conversation with your boss. Find out what concerns him or her most about your being gone for a while, and see if you can figure out how to address those concerns."

Frankel has a few words of advice for anyone who hasn't yet needed to take FMLA leave, but who may have to someday (which, with the aging of the population, might well include most of us, sooner or later). "It's human nature to cut people more slack, and empathize with them more, when we know them well and like them," she says. "So it helps to build warm, collegial relationships at work, and have those loyalties in place, before you need them." Noted.