Is it "OK"? Certainly. There are plenty of homes without a range hood.

Is it a good idea? Absolutely not.

Besides the obvious benefit - removing the smoke & odors - range hoods offer many more perks:

* Airborne grease gets captured = no grease film on cabinets = easier cleanup.
* Air is forced out of the kitchen = pressure drops = fresh air diffuses in from the outside.
* Excess heat is removed = more comfortable cooking experience.

...and so on, and so on...

We have written an article detailing the 10 most important benefits of having a range hood - please feel free to take a look:

http://www.futurofuturo.com/rangehood-FAQ-10-reasons-to-have-a-range-hood.php

P.S. There are plenty of designer range hood models that offer slim profiles with minimal view obstruction. Please feel free to view our selection of 200+ models: http://www.futurofuturo.com/Island-Range-Hoods

P.P.S. A cathedral ceiling is not a "showstopper" - there are several ways to install an island range hood under a sloped / cathedral ceiling:

http://www.futurofuturo.com/rangehood-FAQ-sloped-ceiling.php

Can you tell me how high the range hood should be above the cook top, is 35 inches too high?

Have you considered a down draft hood? Retracts when not in use, it could be a solution for you.

Yes, I am considering this option. Do you know what a good brand is? I'm debating between Jenn Air and Thermodore. Open to other suggestions too.

Hopefully, one of the appliance Pros will lend their expertise on that referral. My rule of thumb for quality appliances; the more technical, the more moving parts, the greater the warranty I look for. My experience is that down-draft ventilation does not perform as effectively as overhead ventilation. The best design to look for in a down-draft unit is one that rises to at least 12" or more above the cooktop surface. When this decision is made, it is often based on aesthetics over performance, so you should be aware of that concession when cooking in the future. All the best in you decision making.

@umpa: we generally recommend placing the range hood 26" - 30" above the counter, for the following reasons:

1.) Capture performance - if the hood is too high, the stream of smoke/steam/grease will spread out too far, and miss the range hood completely.
2.) Usability - it's a good idea to keep the lights *below* eye level, so you're not dazzled every time you need to adjust something on the stove.
3.) Ease of cleaning - having the range hood at/below line of sight also makes it easier to clean.

Thank you all these valuable pointers. I'm still investigating the possibility of doing an overhead vent. Fingures crossed hoping the quote for this will not be too high.