I started to share a contemporary children's rhyme that is very much like the one Cluin posted when it occurred to me that I had posted that rhyme on a thread like this before - thread.cfm?threadid=89534 Origin of 'Chink-a-boo Man'.
Here's the post that I wrote to that thread on March 11, 2006. Note that within the text of that post I refer to a comment that was posted on that thread by Sinsull as well as a comment that was posted on that thread by GUEST,J C. Those posts are also missing, and there are probably additional comments that are missing.

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Subject: RE: Origins: Chink a Boo Man
From: Azizi - PM
Date: 11 Mar 06 - 11:59 AM

Sinsull and others,

I am interested in collecting examples of contemporary children's rhymes. I am also interested in documenting -whenever possible-the sources for those rhymes, and what those rhymes mean to the children reciting them.

I would paraphrase GUEST,J C 's comment that "there [may be] a basic difference in intent between the phrases you quote and those deliberately designed to malign and demean a whole group of people".

It appears to me that alot of times children reciting a rhyme don't know or care what the words mean. They are more "in to" the rhythm of the words and the performance activity {handclapping, jumping rope, foot stomping, etc} that goes with the rhyme.

But as to Sinsull's question would I be interested in collecting a rhyme with the words
"Ching Ching Chinamen
Sittin' on a fence
Trying to make a dollar out fifteen cents."

My answer is yes. My goal is to collect & document rhymes for the historical record, and as a means of gathering insight into the world of children. Therefore, some rhymes that I collect don't conform with my moral standards, but that isn't the point.

That said, I try to include commentary along with the rhyme example so that information is preserved about what the rhyme means/meant to the performers of that rhyme. That's why I believe that demographical information {including age, gender, race and.or ethnicity is important.

I also try to add information about the possible source of the rhyme, and the rhyme's possible cultural meaning [i.e. for "Chink Chink Chinaman" that the rhyme teases Chinese people and other people with slanted eyes].

I'm very interested in how children's rhymes change over time and among different groups of people. For example, in 1997 I collected this handclap rhyme from elementary school aged African American girls & boys in Pittsburgh, PA area that is similar to "Chink Chink Chinaman":

"Shimmy Shimmy China
Sittin on a fence
tryin to make a dollar
outta 65 cents.
He missed. He missed.
He missed like this; like this. like this.

-snip-

In this example "Shimmy Shimmy China" is used as a nickname of a boy or man who is sitting on a fence. Even with the use of the name "China", there is no allusion to Chinese people. And certainly no racial teasing is intended or perceived.

All this to say, that we adults have to be careful not to read our issues into kid's creative works.