Low Hanging Fruit
There are at least three reasons why focusing on low hanging fruit can be a problem for an innovation team. First, low hanging fruit is usually somewhat obvious. If you attack a lot of problems or opportunities that others could have done if they only found the time, then the organization will ask if your initiative is really valuable, since you are only implementing ideas that have already existed. Second, if there is a lot of "low hanging fruit" - and there usually isn't - that says a lot about the organization. What motivated manager leaves a lot of good ideas on the table? There may be some easily identifiable ideas that can be quickly implemented, but I'll suggest to you that you'll run out of those fairly quickly.
Now, if you've been lucky enough to select and implement these simple ideas that have been just lying there for the taking, you've probably created a few enemies along the way, since you've either: 1) demonstrated another manager's lack of insight and/or talent or 2) taken an idea that they were going to pursue. Additionally, if you can implement a lot of ideas that are very simple to identify and bring to market, you've not asked for the firm to commit any resources, and you are probably implementing very incremental ideas.
This last point leads to the problem of expectations. Once you've implemented the easy ones, if there are any, then your pipeline is empty. You've demonstrated the capability of implementing simple ideas but you've not created anything new, nor have you prepared the organization for the challenge of creating and implementing more sophisticated ideas that will take longer and cause more change. In fact, what you've done so far is to demonstrate that innovation is about choosing very easy ideas to implement that don't seem to cost much or change the organization. So, the management team will have some concerns when you come to them, eight to twelve months into the program, and ask for more people and more resources in order to really implement innovation as a consistent capability. You'll not have a choice at that point, since a lot of the low hanging fruit will be gone, or taken off the table by other managers. Yet you've never set an expectation of investment or cost, never built a process or a team.
I think it's important to select a few ideas that are "low hanging fruit" to build credibility early, but you need to set the expectations and understand the strategic intent of the senior executives if you want to create a longer term capability. Any individual or consultant can go into just about any firm and find three or four ideas to implement quickly - that's not really a challenge. The challenge will arise when the "low hanging fruit" is gone, and you have to change the managements' expectations about investment and return.