The answer to this depends a lot on the specifics of your situation.
On one occasion a few years ago, I basically told the other company that I had tight time constraints in my current job and couldn't get away for a three or four hour long interview loop, so I'd like to keep it very tight and then schedule something later if there was mutual interest. That worked reasonably well for them, and I was invited back that evening for another round a couple days later. It also probably reduced some of the potential for wasting time on their end, as either one of us could have cut off discussion after the first round after one or both of us came to the conclusion that it wasn't a good fit.
On the other hand, I don't really see anything wrong with a little white lie about going to run an errand, see the doctor, wait for the cable guy or some other typical visit with a professional that keeps the same kind of hours your day job normally requires. This gives you plausible deniability for up to a few hours out of your day.
More recently, I've enjoyed enough professional clout that I can schedule a quick phone call, meet someone from the other firm over a series of coffee or lunch meetings, things I'd be doing anyway at least on occasion with coworkers or friends who work nearby, and both sides can gauge whether we're meant for each other or not on a quick but manageable timeline.
There's really no convenient time for a job interview when you're gainfully employed, and many firms accidentally or intentionally rank people who don't need them more highly than people who are out of work, so it's actually really not a terrible problem to have; most companies are thrilled to work around the schedule of someone they suspect may be a perfect fit.
In terms of your own current work, as long as you have a plausible explanation for a brief absence and, if you are on a tight schedule, you make up for lost time, I wouldn't worry about hurting anyone's feelings while you're meeting with the firm that's pitching you a competing opportunity.
The dynamics change a bit if you're interviewing for a gig that's out of town, but you can adjust to those situations too; I've scheduled phone calls and videoconferences off hours or during lunch before determining whether it was worth the trip to meet in person. Most companies are ok with these arrangements because it's far too expensive to fly you out if you're not really interested. If you actually go out of town to meet the other firm, you schedule it around a weekend so that it doesn't need any explanation other than you're flying out of town for a quick visit with family or friends.
It's possible you may not feel like you're so powerful as to be in control of a job negotiation, so there's a chance you may feel like my advice is meant for someone of unusual professional status, but don't fall into this trap. By being gainfully employed, you are in a position of power with both your present and potential future employer, because you have the luxury of choice. If you get a competing offer, you are suddenly in a position to have two firms make a judgment on your value to them, and then you get to think about whether you're going to learn and develop more at one firm or the other. This is an incredibly nice position to be in.