Is there a more general term for "googling" that doesn't imply a particular search engine?

Nowadays lots of people use the word "google" as a verb, past tense "googled", which generally means to search the web using Google's search engine. When used as a gerund the word is typically "googling". (I'm not sure if these words should be capitalized, but that's not relevant to my question.)

Is there a common word or phrase that can be used in the same way, but that doesn't imply using any particular search engine? I'm curious because not everyone I talk to uses Google.

Example sentence, as a verb:

I _______ for the words "cute kitty" and found lots of results.

Example sentence, as a gerund:

I did some _______ but couldn't find anything relevant.

The best I can think of is "web search" or "web searching" but it strikes me as awkward. I'm hoping there's something a bit better and more natural sounding, especially if it's a single word.

To search [...] online would be my choice. In your example sentence it would read:

I searched for the words "cute kitty" online and found lots of results.

...or...

I searched online for the words "cute kitty" and found lots of results.

As for your other sentence:

I did some searching online but couldn't find anything relevant.

According to Google Ngrams it seems to correlate well with "Google", though I'm quite surprised at those earlier results for the latter! (Perhaps something coming astray during the transcribing...). It also gets a lot more use than "search the net", "search the web" or "search the internet".

Actually to google is becoming a general term to refer to any search engine:

verb (used without object)

• (often lowercase) to use a search engine such as Google to find information, a website address, etc., on the Internet.

(Dictionary.com)

To google:

• As a result of the increasing popularity and dominance of the Google search engine,1 usage of the transitive verb to google (also spelled Google) grew ubiquitously. The neologism commonly refers to searching for information on the World Wide Web, regardless of which search engine is used. The American Dialect Society chose it as the "most useful word of 2002." It was added to the Oxford English Dictionary on June 15, 2006, and to the eleventh edition of the Merriam-Webster Collegiate Dictionary in July 2006.

(Wikipedia)

In the days of the web before Google existed (mid to late '90s), we just called it "searching" or "performing a web search" (web search would be the common noun). Doing so usually entailed using 3 or 4 different search engines to find what you were looking for, as no particular search engine was very good, and each would give different results. Some I remember using include Yahoo!, Alta Vista, Web Crawler, Excite, Lycos, and Ask Jeeves. None were overwhelmingly dominant.

Google was superior enough to the competition at the time it came out that most people abandoned other search engines. It could usually get you what you wanted on the first try. That's when "googling" became the generic verb for searching the web.

As in:

What did your search reveal?

On Wikipedia there is a whole subsection of SEARCHING devoted to Computing Technology.

Computing technology

• Search algorithm, including keyword search
• Search engine technology, software for finding information
  • Web search engine, a service for finding information on the World Wide Web
  • Enterprise search, software or services for finding information within organizations
• Search and optimization for problem solving in Artificial Intelligence

I appears when using search in the proper context it will be known that you used a search 'engine' to peruse the internet for information.