It's hard to answer the question without seeing more of the design scheme, but I'll give you the guideline anyway.
Your question is about a problem of affordance. Affordance is a visual cue that implies action. For example: a thin slot in a soda machine indicates the place in which the user should insert their coin, and the pull lever in a car's door implies that the user should pull the door to open it. Removing these visual cues can confuse the user regarding the way the instrument should be used.
In web design, we have some grounded conventions for how elements should look like. In case of a text field, the convention is a grey-bordered box with a white background. Removing one of these or more of these elements decreases the field's affordance. I predict that it will be harder for users to recognize it's a text field, and therefore create confusion and frustration. I believe that your designer knows that as well, and for that reason suggested other added cues (the flashing cursor).
But don't take my word for it. Make a quick-and-cheap usability test. Go to 3 people who weren't involved in the design of this website, show them a screenshot of the current design, and ask them to "perform a search". Don't just notice if they recognize the search field or not. Pay attention to how long it takes them, what they're saying while they're looking for it ("where is that god d*** search field?!") and how they react when they finally find it (or not!). If you get the sense that they're struggling, go back to your designer and ask him to make the field a bit more conventional.
*An exception would be if this website is trying to convey a message of innovation and "breaking the rules" (e.g. an interactive studio's website). If that's the case, and other elements are also non-conventional on-purpose, then you should stick to the unconventional text field as well.