Though as some other posters have noted, some Russians may use dialect case forms, anyone who is out of diapers uses the full case system. Case is a core concept of the language. The very idea that using cases is a burden is alien to Russian.
If you hear someone speaking Russian while ignoring case and gender, he isn't uneducated, he is a foreigner. He doesn't know it, but nobody is quite sure what he is saying.
In English we have the literary language with fairly strict grammar and a relaxed conversational form with loose grammar. For example, the literary form is "It is I." wereas the conversational form is "It's me." In the conversational example we use the wrong case. We can do this because case is now so weak a concept in English that nobody notices that the meaning which "me" as opposed to "I" once conveyed does not fit logically into the sentence.
As a result, English speakers learning Russian have a very hard time accepting that case is an important concept. They assume that it, as in English, is an affectation of the educated which they can safely ignore. Nothing could be furthur from the truth. If you say "It's me." (Это меня.) in Russian people will look at you blankly.
I have had some success explaining this to Americans using the following example: In popular culture Tarzan says "Me Tarzan!". We understand this to mean "I am Tarzan." This is because our ideas as to the difference between "I" and "me" are hazy. Also, we expect the subject of the sentence to come first. So we accept "me" as the subject even though the fact that it is in the objective case theoretically indicates that it is either the direct or indirect object.
But a Russian does not expect the subject to come first and very clearly understands the difference betwee "я" (I) and "меня" (me). He will immediately reach two firm conclusions: 1) the speaker is most definitely not Tarzan, and 2) the sentence is incomplete. The part which tells what Tarzan did to the speaker is missing. The sentence would be complete if we added a verb such as "saw".
So no, a Russian would not consider a form of the language in which the use of cases was relaxed to be simpler. He would probably consider it to be borderline incomprehensible baby talk.
You mention a difference in case use between classical Latin and vulgar Latin. I am not well informed about this subject, but some things I have read have left the impression that classical Latin reflects an earlier form of the language before so many cases were lost. There is a general trend for languages to lose cases due to pronunciation shifts or foreign influence.
EDIT to address comments
As Andrey Chernakhovskiy and @Annix point out "Это меня!" is a valid utterance if a subject and verb are supplied by the context. The difference between English and Russian is this: If an American says "It is me.", the grammatical error is not even noticed because case plays almost no role in conveying meaning. But if our American is learning Russian and translates it word-for-word and says "Это меня!", he is likely to get blank stares because his use of case implies that there are a subject and verb in the context when there are none.
@Annix compares the ommision of case to the ommision of the word "of" in English. This is a brilliant example. I wish I had thought of it.
@SixthOfFour asked how Tarzan might speak while still learning Russian. In reality his speach would be incomprehensible much of the time. But in a movie it would contain errors carefully selected to impair comprehensibility as little as possible. In Russian he might leave nouns undeclined, avoid the use of pronouns, and speak in the third person. As @Annix points out Tarzan could utter the words "Меня Тарзан!" in a context which implies "зовут", but in my experience beginning speakers do not understand grammar sufficently to understand that this is possible.
EDIT to address points raised by @dainichi
@dainichi comments served to further illuminate the differences between how English and Russian speakers perceive case.
@dainichi objected to my description of "It's me" as "wrong" according to the rules of formal literary English. Whether we approve or disapprove of the prescriptive approach, the question that was asked is whether ordinary Russians and Germans use cases and conjugate verbs as the prescriptive rules in grammars require or are these rules an affectation of over-educated snobs.
As an example I referred to the well known prescriptive rule for English and Russian which states that the predicate nominative must be in the nominative case. I then cited a well-accepted expression ("It's me") which violates this rule.
My point was that for English it is reasonable to discuss whether and when we should follow this conservative prescriptive rule seeing as it is based on arbitrary or outdated concepts of case. But if we suggested to even the most uneducated Russian that in informal speech "Это меня" (It's me) is an acceptable substitute for "Это я" (It's I), he would think we were either joking or very confused. It is not that he knows and loves prescriptive rules. It is because he is acutely aware of the message about the role of the noun in the sentence which each case conveys.
If we English speakers were to ascribed this level of meaning to case, we would not be able to say "It's me." or "Me and my friend went to the park." because these utterances would appear to be nonsensical. Why “It's me.” is acceptable is an interesting question. Maybe as @dainichi suggests we now perceive “me” as nominative or we have adopted French or Danish practice. Or maybe we just got confused. But no matter what the reason saying "It's me." would have been practically impossible had we perceived cases as Russians do. This is why even uneducated Russians use the full case system.