(Crossposted from your question in /r/AskAnthropology)
Like a lot of Gimbutas' work, there are some elements of truth to it but on the whole it's highly speculative and not accepted by the majority of prehistorians.
There are a lot of similarities between the various Neolithic cultures of Southeastern Europe. Between 5,000 and 3,500 BCE, the whole area between the Danube Valley and the Dniepr was covered in prosperous agricultural villages. They all seem to have been largely egalitarian, with an emphasis on the domestic life of single families. They were adept and experimental craftspeople, especially in terms of ceramic technology, and in the latter part they were amongst the first in the world to invent metalworking. They had some shared rituals, including making abstract, feminine figurines and deliberately burning houses at their end of their life. All these cultures grew out of the first farmers to colonise the region in the 6th millennium BCE, and all of them appear to have been radically transformed at the end of the fourth millennium, when village life gave way to a more mobile, pastoral existence and the first signs of inequality and hierarchy begin to be seen.
But with that said, most prehistorians would say these societies were far too diffuse and diverse to be called a unified civilization, as Gimbutas insisted. It's more likely to come from their shared origin in the region's first farming cultures and continued cultural links. Moreover, there is very little evidence for what she asserted were key elements of "Old Europe": matrifocality, goddess-worship, pacifism, a shared language, or proto-writing. Her conception of Old Europe was also very tied up with the polar-opposite entity she thought destroyed it: the patriarchal, warlike, horse-nomad Kurgan culture. But that's another one of her ideas that didn't really catch on in mainstream archaeology.
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