Work on making people feel safe with the situation before asking for direct feedback.
A few examples of how this could be done:
Give them more than a few weeks to get comfortable with you and the company culture.
Give concrete examples of things that have improved due to employee feedback.
It's hard to offer better advice through a HN post but easy to observe in person. Perhaps you can hire an external consultant to help or ask a mentor or advisor to fill this role by spending a few days in your office.
We're a small (45 people) software engineering and UX consultancy with offices in Glasgow and London. We work in a variety of sectors but we've begun to focus on the Galleries, Libraries, Archives, and Museums (GLAM) sector in particular.
In addition to consulting we're building our own product call the digital libraries cloud services (DLCS), which makes use of a number of new standards such as IIF and open web annotations.
The interview process involves an initial phone screen, a broad quantitive technical test and a final qualitative "culture fit" interview. Time and availability permitting the 2nd and 3rd step can be done in a single visit.
These are all valid points but many organisations are not at the level of maturity where the inputs to the engineering team are of the high quality you're describing.
Additionally there's an element of negotiation or dialogue around requirements which can often lead to cheaper/quicker to market/better solutions with input from engineering.