Not really, no. While the migrations of the fourth and fifth centuries certainly brought about important changes in the social and political landscape of the late ancient Roman world, the groups that entered the Empire were overwhelmingly in the minority. The Visigoths, who sacked Rome (410) and carved out a kingdom in southern France and Spain, likely only numbered in the lows tens of thousands during this early period. Even the old estimates of 100,000 individuals --a number that has largely been discarded-- would still be dwarfed by the native population of Iberian peninsula. Wholesale population replacement in the early medieval world was a difficult feat to achieve, and generally speaking, was not the aim of these migratory groups. Remember, these groups were not looking to destroy the Roman Empire, but to take advantage of its wealth, prosperity, and prestige.
The same general trend tends to hold for the other 'barbarian' successor states: the Franks in Gaul, the Burgundians in the Loire Valley, the Ostrogoths in Northern Italy, the Angles, Saxons, and Jutes in England, though I know less about these other groups. If you're interested in reading more, I have answered similar questions about the Visigoths and early medieval Spain on /r/AskHistorians previously:
From a socio-cultural stand point, however, one could argue I guess that the origins of European 'nations' have their roots in this period. Or rather, nationalist historians have looked back to this period to find the beginnings of their respective peoples, and in doing so have created narratives that tied present identities to the past. So in a sense, modern European nations are 'heirs' to the peoples of this period, even if their genetic imprint on the population is minimal.
The other people responding no have no sense of the scope of genealogy. In reality it is statistically impossible for a typical "Western person" such as you describe to avoid being descended from these Germanic tribes. A person's family would have to live in absolute isolation for generations for this to be possible.
If you compare the number of theoretical ancestors a person has alive (i.e. 2 to the power of the number of generations) in a particular generation with the number of people alive during the generations these tribes lived you are faced with the fact that everyone has a great many duplicate ancestors (i.e. inbreeding). Just going back to the year 1000 requires over a billion unique ancestors, many times more people than lived on the Earth at the time.