I don't think Islam itself is in any way a problem. My personal issues with religion always have come from the prerequisite of organization.
The issue I personally have diagnosed in discussions about the Islam is that its often compared to Christianity almost in full, but there are fundamental differences in the application of the faith (this is often in conversation in regards to extremism. There are definitely branches of islam out there that adhere to a very different set of values, and some that are a lot more like modern day Christianity.) The term Christianity is also problematic in those discussion, because it's a loaded term with tons of history. It puts the Islam in a tradition of abrahamic faiths, and roots in with Christianity. That's true, no dismissing that. The problem is that the Christianity of 6th century Earth is not the modern form. But not being like Christianity is not a problem for a religion at all. I just want to point out that the comparison often feels wrong to me. It's also often discussed as if its one giant pillar of faith, which is also an issue. But people are less educated about the Islam in the West, most often, and it does happen with a certain regularity when discussing Christianity as well.
My views on it is that any religion is in constant need of modernization. The world changes a lot faster than religion is prepared for, so beliefs carried over from generation to generation are quickly outdated. Islam is not immune, but neither is Christianity, or Hinduism. A religion always needs to be critical of both the world and itself. Not every change the world goes through is ideal, but it is important to realize your own beliefs and their shortcomings.
I think this is very much at the core of why I'm not a fan of organized religion, but prefer a personalized form of belief. Because it asks you as a person to constantly reassess, and put your own discoveries in the context about the world with your preconceived notions. It's a lot harder to do if there's a mandated text and a community of pressure.
I do have to say that I'm often a bit baffled by this question. It implies the unmodern beliefs of some Muslims are not just present, but rooted so firmly in the religion that they're inseparable. Christianity has a place in modern society, homophobia doesn't. That's possible because the hatred of the other is not central to Christianity. I want to treat any religion that way. Fundamentalist religious terrorism doesn't have a place in modern society. Don't equate that with Islam. If its a symptom of certain denominations, then those don't have a part in modern society. But not the entirety. That's disregarding the parts, the many streams that flow from a river with one name.