I'd have to answer this from an employer's point of view: Why?
It's such a misnomer that as an employee you have to hide things - that it's you against the company when you decide to leave. I think that's a real misunderstanding of the relationship companies have with their employees and vice versa. I also think it's driven predominantly from fear of being terminated and putting yourself in a financial situation you don't want to deal with (for example, the employer gets pissed and fires you for being a traitor).
The reality is that employers appreciate the ability to plan, strategize and stay ahead of obstacles. If you throw them a surprise by quitting "out of the blue" it's a real drag on the entrepreneur.
It's also completely unnecessary.
It's so much better to be open and honest and explain the situation to the people who have worked hard to provide you a salary, benefits and an environment where you could grow. You never know, the employer might even address your unhappiness and you could decide to stay.
If you cloak and dagger the exercise, you may burn a bridge with your existing employer, especially if you were in the midst of a project and don't give adequate heads up time. If you hit it straight up like a seasoned professional and tell your employer, you'll most likely garner respect and you can work through a transition plan such that everybody wins (or a plan to keep you and make you happy).
Besides, your employer will eventually find out anyhow because your behavior will change, your resume will be floating around and people in the office will talk. All that behind the scenes activity just sheds a bad light on you.
And if, in the end, the employer acts like a jack-a$$ and terminates you on the spot, it's my opinion that leadership, living by example and doing the right thing sometimes comes at a cost.
Besides, the number of times positive things come out of you living your life that way outweighs the one-off jack-a$$e$ in life.
@Chris makes a good point about being open.
But I can see from your perspective, why someone would like to keep this a secret: to not potentially jeopardize your current job in case another opportunity doesn't pan out.
What I'd recommend is to group multiple interviews and calls on a single day per week (instead of spread out over the week). This way you won't raise any alarms if you were out for a day... and still your productive self at work the other days.